Vincent Lafforgue
The report below describes the applications of Banach KK-theory to a conjecture of P. Baum and A. Connes about the K-theory of group $C^*$-algebras, and a new proof of the classification by Harish-Chandra, the construction by Parthasarathy and the exhaustion by Atiyah and Schmid of the discrete series representations of connected semi-simple Lie groups.
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We prove the Baum-Connes conjecture for hyperbolic groups and their subgroups.
This paper gives a proof of the Baum-Connes conjecture with coefficients for hyperbolic groups. More precisely the injectivity of the Baum-Connes map was established by Kasparov and Skandalis and we prove the surjectivity.
Let G be a locally compact group and rho a non-unitary finite dimensional representation of G. We consider tensor products of rho by some unitary representations of G in order to define two Banach algebras analogous to the group C*-algebras, C*(G) and C*_r(G). We calculate the K-theory of such algebras for a large class of groups satisfying the Baum-Connes conjecture.
We introduce property $(T_{Schur},G,K)$ and prove it for some non-cocompact lattice in $Sp_4$ in a local field of finite characteristic. We show that property $(T_{Schur},G,K)$ for a non-cocompact lattice $Γ$ in a higher rank almost simple algebraic group in a local field is an obstacle to proving the Baum Connes conjecture without coefficient for $Γ$ with known methods, and this is stronger than the well-known fact that $Γ$ does not have the property of rapid decay (property (RD)). It is the first example (as announced in [Laf10a]) for which all known methods for proving the Baum Connes conje…
The Baum-Connes conjecture: a concise course
本文提供了一门10小时课程的讲义，系统介绍Baum-Connes猜想，涵盖其背景、证明思路与相关非交换几何工具。
We define and compare two bivariant generalizations of the topological $K$-group $K^\top(G)$ for a topological group $G$. We consider the Baum-Connes conjecture in this context and study its relation to the usual Baum-Connes conjecture.
We associate a non-commutative $C^*$-algebra with any locally finite simplicial complex. We determine the $K$-theory of these algebras and show that they can be used to obtain a conceptual explanation for the Baum-Connes conjecture.
On the quantitative coarse Baum-Connes conjecture with coefficients
本文提出了针对真度量空间的带系数定量粗Baum-Connes猜想（QCBC），证明其可由带系数的粗Baum-Connes猜想导出，并指出QCBC最终可归结为一列有界度量空间的一致定量猜想。
We establish the Hasse principle (local-global principle) in the context of the Baum-Connes conjecture with coefficients. We illustrate this principle with the discrete group $GL(2,F)$ where $F$ is any global field.
The equivariant coarse Baum-Connes conjecture was firstly introduced by Roe [29] as a unified way to approach both the Baum-Connes conjecture and its coarse counterpart. In this paper, we prove that if an a-T-menable group $Γ$ acts properly and isometrically on a bounded geometry metric space $X$ with controlled distortion such that the quotient space $X/Γ$ is coarsely embeddable, then the equivariant coarse Baum-Connes conjecture holds for this action. This answers a question posed in [8] affirmatively.
Given a (not necessarily discrete) proper metric space $M$ with bounded geometry, we define a groupoid $G(M)$. We show that the coarse Baum--Connes conjecture with coefficients, which states that the assembly map with coefficients for G(M) is an isomorphism, is hereditary by taking closed subspaces.
We show that the Baum-Connes morphism twisted by a non-unitary representation, defined in [GA08], is an isomorphism for a large class of groups satisfying the Baum-Connes conjecture. Such class contains all the real semi-simple Lie groups, all hyperbolic groups and many infinite discret groups having Kazhdan's property (T). We define a tensorisation by a non-unitary finite dimensional representation on the left handside of the Baum-Connes morphism and we show that its analogue in K-theory must be defined on the K-theory of the twisted group algebras introduced in [GA07b].
The Baum-Connes map for finitely generated free abelian groups is a K-theoretic analogue of the Fourier-Mukai transform from algebraic geometry. We describe this K-theoretic transform in the language of topological correspondences, and compute its action on K-theory (of tori) described geometrically in terms of Baum-Douglas cocycles, showing that the Fourier-Mukai transform maps the class of a subtorus to the class of a suitably defined dual torus. We deduce the Fourier-Mukai inversion formula. We use these results to give a purely geometric description of the Baum-Connes assembly map for free…
We introduce a new method for studying the Baum-Connes conjecture, which we call the direct splitting method. The method can simplify and clarify proofs of some of the known cases of the conjecture. In a separate paper, with J. Brodzki, E. Guentner and N. Higson, a similar idea will be used to give a finite-dimensional proof of the Baum-Connes conjecture for groups which act properly and co-compactly on a finite-dimensional CAT(0)-cubical space.
In this paper, we verify the $L^p$ coarse Baum-Connes conjecture for spaces with finite asymptotic dimension for $p\in[1,\infty)$. We also show that the $K$-theory of $L^p$ Roe algebras are independent of $p\in(1,\infty)$ for spaces with finite asymptotic dimension.
In this paper, we connect the rigidity problem and the coarse Baum-Connes conjecture for Roe algebras. In particular, we show that if $X$ and $Y$ are two uniformly locally finite metric spaces such that their Roe algebras are $*$-isomorphic, then $X$ and $Y$ are coarsely equivalent provided either $X$ or $Y$ satisfies the coarse Baum-Connes conjecture with coefficients. It is well-known that coarse embeddability into a Hilbert space implies the coarse Baum-Connes conjecture with coefficients. On the other hand, we provide a new example of a finitely generated group satisfying the coarse Baum-C…
The Baum-Connes conjecture predicts that a certain assembly map is an isomorphism. We identify the homotopy theoretical construction of the assembly map by Davis and Lück with the category theoretical construction by Meyer and Nest. This extends the result of Hambleton and Pedersen to arbitrary coefficients. Our approach uses abstract properties rather than explicit constructions and is formally similar to Meyer's and Nest's identification of their assembly map with the original construction of the assembly map by Baum, Connes and Higson.
The coarse Baum-Connes conjecture with filtered coefficients and product metric spaces
本文受定量K-理论启发，提出带滤过系数的粗Baum-Connes猜想，推广原始猜想，并证明其满足类在积下封闭，进而发现新的积度量空间例子。
$\ell^p$-coarse Baum-Connes conjecture for $\ell^{q}$-coarse embeddable spaces
证明了可粗嵌入$\ell^q$空间的空间满足$\ell^p$粗Baum-Connes猜想，覆盖所有$p,q\in[1,\infty)$。
We give a decomposition of the equivariant Kasparov category for discrete quantum group with torsions. As an outcome, we show that the crossed product by a discrete quantum group in a certain class preserves the UCT. We then show that quasidiagonality of a reduced C*-algebra of a countable discrete quantum group $Γ$ implies that $Γ$ is amenable, and deduce from the work of Tikuisis, White and Winter, and the results in the first part of the paper, the converse (i.e. the quantum Rosenberg Conjecture) for a large class of countable discrete unimodular quantum groups. We also note that the unimod…
We present a history of the Baum-Connes conjecture, the methods involved, the current status, and the mathematics it generated.
We introduce a new variant of the coarse Baum-Connes conjecture designed to tackle coarsely disconnected metric spaces called the boundary coarse Baum-Connes conjecture. We prove this conjecture for many coarsely disconnected spaces that are known to be counterexamples to the coarse Baum-Connes conjecture. In particular, we give a geometric proof of this conjecture for spaces of graphs that have large girth and bounded vertex degree. We then connect the boundary conjecture to the coarse Baum-Connes conjecture using homological methods, which allows us to exhibit all the current uniformly discr…
We construct a Baum--Connes assembly map localised at the unit element of a discrete group $Γ$. This morphism, called $μ_τ$, is defined in $KK$-theory with coefficients in $\mathbb{R}$ by means of the action of the projection $[τ]\in KK_{\mathbb{R}}^Γ(\mathbb{C},\mathbb{C})$ canonically associated to the group trace of $Γ$. We show that the corresponding $τ$-Baum--Connes conjecture is weaker then the classical one but still implies the strong Novikov conjecture. The right hand side of $μ_τ$ is functorial with respect to the group $Γ$.
We study a Going-Down (or restriction) principle for ample groupoids and its applications. The Going-Down principle for locally compact groups was developed by Chabert, Echterhoff and Oyono-Oyono and allows to study certain functors, that arise in the context of the topological K-theory of a locally compact group, in terms of their restrictions to compact subgroups. We extend this principle to the class of ample Hausdorff groupoids using Le Gall's groupoid equivariant version of Kasparov's bivariant KK-theory. Moreover, we provide an application to the Baum-Connes conjecture for ample groupoid…
We show that the real Baum-Connes conjecture for abelian groups, possibly twisted by a cocycle, explains the isomorphisms of (twisted) KR-groups that underlie all T-dualities of torus orientifold string theories.
In this paper we consider the Baum-Connes correspondence for the affine and extended affine Weyl groups of a compact connected semisimple Lie group. We show that the Baum-Connes correspondence in this context arises from Langlands duality for the Lie group.
We reformulate the Baum-Connes conjecture with coefficients by introducing a new crossed product functor for C*-algebras. All confirming examples for the original Baum-Connes conjecture remain confirming examples for the reformulated conjecture, and at present there are no known counterexamples to the reformulated conjecture. Moreover, some of the known expander-based counterexamples to the original Baum-Connes conjecture become confirming examples for our reformulated conjecture.
We introduce the notion of proper Kasparov cycles for Kasparov's G-equivariant KK-theory for a general locally compact, second countable topological group G. We show that for any proper Kasparov cycle, its induced map on K-theory factors through the left-hand side of the Baum-Connes conjecture. This allows us to upgrade the direct splitting method, a recent new approach to the Baum-Connes conjecture which, in contrast to the standard gamma element method (the Dirac dual-Dirac method), avoids the need of constructing proper algebras and the Dirac and the dual-Dirac elements. We introduce the no…
The quantitative coarse Baum-Connes conjecture for free products
本文在自由积G*H中证明了定量粗 Baum-Connes 猜想，前提是该猜想在G和H上成立。
Coarse Baum-Connes and warped cones: failure of surjectivity in odd degree
本研究构造统一扭曲锥证明了Roe的猜想，证实粗Baum-Connes猜想在奇次度下满射性失效，且阻碍来源与Roe的预期不同。
We prove an analogue of the Baum-Connes conjecture for free orthogonal quantum groups. More precisely, we show that these quantum groups have a $ γ$-element and that $ γ= 1 $. It follows that free orthogonal quantum groups are $ K $-amenable. We compute explicitly their $ K $-theory and deduce in the unimodular case that the corresponding reduced $ C^* $-algebras do not contain nontrivial idempotents. Our approach is based on the reformulation of the Baum-Connes conjecture by Meyer and Nest using the language of triangulated categories. An important ingredient is the theory of monoidal equival…
We prove that if the classical Baum-Connes conjecture in complex K-theory is true (for a given discrete group G), then the conjecture is also true in the real case (for the same group G). The essential ingredients of the proof are the descent theorem in topological K-theory (math.KT/0509396) and a Paschke duality for C*-algebras proved by John Roe (Q.J. Math. 55 (2004) 325-331). According to S. Stolz, this result is linked with the existence of a Riemannian metric of positive scalar curvature on a compact connected spin manifold with G as fundamental group.
We give a new proof of the Baum--Connes conjecture with coefficients for any second countable, locally compact topological group that acts properly and cocompactly on a finite-dimensional CAT(0)-cubical space with bounded geometry. The proof uses the Julg-Valette complex of a CAT(0)-cubical space introduced by the first three authors, and the direct splitting method in Kasparov theory developed by the last author.
Coarse geometry of Hecke pairs and the Baum-Connes conjecture
本研究借助Hecke对陪集空间与Schlichting完备化的粗几何，证明了余Haagerup性质下Baum-Connes和Novikov猜想的新稳定性结果，推广了既有结论并给出满足带系数Baum-Connes猜想的S-算术群新实例。
We introduce higher-dimensional analogs of Kazhdan projections in matrix algebras over group $C^*$-algebras and Roe algebras. These projections are constructed in the framework of cohomology with coefficients in unitary representations and in certain cases give rise to non-trivial $K$-theory classes. We apply the higher Kazhdan projections to establish a relation between $\ell_2$-Betti numbers of a group and surjectivity of different Baum-Connes type assembly maps.
Expanders are counterexamples to the $\ell^p$ coarse Baum-Connes conjecture
本文通过构造来自剩余有限双曲群的膨胀图，证明了 $\ell^p$ 粗 Baum-Connes 装配映射在 $1<p<\infty$ 时不是满射，从而给出该猜想的反例。
The well known "associativity property" of the crossed product by a semi-direct product of discrete groups is generalized into the context of discrete \emph{quantum} groups. This decomposition allows to define an appropriate triangulated functor relating the Baum-Connes property for the quantum semi-direct product to the Baum-Connes property for the discrete quantum groups involved in the construction. The corresponding stability result for the Baum-Connes property generalizes a result of J. Chabert for a quantum semi-direct product under torsion-freeness assumption. The $K$-amenability connex…
We introduce a notion of fibred coarse embedding into Hilbert space for metric spaces, which is a generalization of Gromov's notion of coarse embedding into Hilbert space. It turns out that a large class of expander graphs admit such an embedding. We show that the maximal coarse Baum-Connes conjecture holds for metric spaces with bounded geometry which admit a fibred coarse embedding into Hilbert space.
Using the unbounded picture of analytical K-homology, we associate a well-defined K-homology class to an unbounded symmetric operator satisfying certain mild technical conditions. We also establish an ``addition formula'' for the Dirac operator on the circle and for the Dolbeault operator on closed surfaces. Two proofs are provided, one using topology and the other one, surprisingly involved, sticking to analysis, on the basis of the previous result. As a second application, we construct, in a purely analytical language, various homomorphisms linking the homology of a group in low degree, the …
$L^p$ coarse Baum-Connes conjecture via $C_{0}$ coarse geometry
本文通过 $C_0$ 粗结构研究 $L^p$ 粗 Baum-Contes 猜想，证明有限维simplicial复形在统一球面度量下成立，并构造障碍群，证明有限渐近维数下障碍群消失，从而提供新的证明。
The twisted coarse Baum--Connes conjecture and relative hyperbolic groups
本文引入稳定粗代数并提出扭曲粗Baum--Connes猜想，证明其在粗等价等操作下保持，并应用于相对双曲群，得出群满足猜想当且仅当其每个双曲子群满足。
We study general properties of exotic crossed-product functors and characterise those which extend to functors on equivariant C*-algebra categories based on correspondences. We show that every such functor allows the construction of a descent in KK-theory and we use this to show that all crossed products by correspondence functors of K-amenable groups are KK-equivalent. We also show that for second countable groups the minimal exact Morita compatible crossed-product functor used in the new formulation of the Baum-Connes conjecture by Baum, Guentner and Willett extends to correspondences when r…
Dehn fillings, equivariant homology, and the Baum-Connes conjecture
本文建立 Cohen-Lyndon 三元组与等变同调的联系，证明有限圈积系数的 Baum-Connes 猜想对离散双曲群及一大类相对双曲群（含实秩一简单 Lie 群中的格）成立。
Controlled Analytic Properties and the Quantitative Baum-Connes Conjecture
研究显示，对于一类空隙超几何群，巴乌-康纳斯装配映射在量化上是同构的，并解释了该类群包含许多具有大环长的图序列的群，因此不具有性质（A）。
A groupoid approach to the equivariant coarse Baum--Connes conjecture
本文通过群体群论方法研究等价粗 Baum--Connes 猜想，引入等价粗群体群并证明其与等价粗 Baum--Connes 猜想的等价性，进一步证明了当空间 X 可以粗嵌入到 Hilbert 空间时，等价粗 Novikov 猜想成立。
The Unitary Conjugation Groupoid as a Universal Mediator of the Baum--Connes Assembly Map
本文展示单位ary共轭群oid作为 Baum--Connes 装配映射的通用中介，扩展了 K_1 指数框架到非 Type I 情况，并通过 Morita 等价将单位ary 共轭群oid与变换群oid联系起来。
A Baum-Connes assembly map for essential semigroup crossed products
构造了可分离C*-代数上逆半群Fell丛的等变E-理论和Baum-Connes装配映射，推广了先前工作，并建立了相关函子性质。
The maximal coarse Baum-Connes conjecture for spaces that admit an A-by-FCE coarse fibration structure
该研究提出度量空间的A-by-FCE粗纤维化结构，推广了已有群扩张的A-by-CE结构，证明了有界几何下该类空间满足极大粗Baum-Connes猜想，并验证了相对扩张子、盒空间等非FCE空间也满足该猜想。
The goal of this article is to provide a bridge between the gamma element method for the Baum--Connes conjecture (the Dirac dual-Dirac method) and the controlled algebraic approach of Roe and Yu (localization algebras). For any second countable, locally compact group G, we study the reduced crossed product algebras of the representable localization algebras for proper G-spaces. We show that the naturally defined forget-control map is equivalent to the Baum--Connes assembly map for any locally compact group G and for any coefficient G-C*-algebra B. We describe the gamma element method for the B…
We define and study an analogue of the Baum-Connes assembly map for complex semisimple quantum groups, that is, Drinfeld doubles of $ q $-deformations of compact semisimple Lie groups. Our starting point is the deformation picture of the Baum-Connes assembly map for a complex semisimple Lie group $ G $, which allows one to express the $ K $-theory of the reduced group $ C^* $-algebra of $ G $ in terms of the $ K $-theory of its associated Cartan motion group. The latter can be identified with the semidirect product of the maximal compact subgroup $ K $ acting on $ \mathfrak{k}^* $ via the coad…
A localization algebra approach to the Baum-Connes conjecture for extensions
针对离散可数群扩展的Baum-Connes系数猜想，已知当商群及商群有限子群原像满足猜想时群本身满足，本文用局部化代数方法证明该结果。
The coarse Baum-Connes conjecture for certain relative expanders
本文证明了若有限群正合列序列中核与商群的粗不相交并可粗嵌入Hilbert空间，则原群的粗不相交并满足粗Baum-Connes猜想，由此解决了相对膨胀图相关的公开问题。
A going-down principle for {é}tale groupoids and the Baum-Connes conjecture
本文扩展了étale群胚的下行原理，通过群胚单纯复形研究证明了一般étale群胚的Baum-Connes装配映射分裂单射，并应用于拓扑K-理论连续性和Künneth公式。
Categorical approach to the Baum-Connes conjecture for étale groupoids
本文利用等变Kasparov范畴的三角化结构研究étale groupoid的Baum-Connes猜想，证明弱可缩对象与projective对象子范畴互补，提出强Baum-Connes猜想并给出多个推论及应用。
We study in this paper the maximal version of the coarse Baum-Connes assembly map for families of expanding graphs arising from residually finite groups. Unlike for the usual Roe algebra, we show that this assembly map is closely related to the (maximal) Baum-Connes assembly map for the group and is an isomorphism for a class of expanders. We also introduce a quantitative Baum-Connes assembly map and discuss its connections to K-theory of (maximal) Roe algebras.
We formulate a version of Baum-Connes' conjecture for a discrete quantum group, building on our earlier work (\cite{GK}). Given such a quantum group $\cla$, we construct a directed family $\{\cle_F \}$ of $C^*$-algebras ($F$ varying over some suitable index set), borrowing the ideas of \cite{cuntz}, such that there is a natural action of $\cla$ on each $\cle_F$ satisfying the assumptions of \cite{GK}, which makes it possible to define the "analytical assembly map", say $μ^{r,F}_i$, $i=0,1,$ as in \cite{GK}, from the $\cla$-equivariant $K$-homolgy groups of $\cle_F$ to the $K$-theory groups of …
Twisted Roe algebras and their $K$-theory
本文提出扭曲Roe代数与带系数的扭曲粗Baum-Connes猜想，证明在粗纤维化结构下该猜想可由底空间与纤维传递，并应用于粗嵌入群扩张，得出粗Baum-Connes猜想成立。
Sharp mapping properties of Poisson transforms and the Baum-Connes conjecture
本文证明了Poisson变换在分布层上精确的量化类比，展示了其将Sobolev空间有界映射到L^2空间，并证明了与Furstenberg紧化上光滑函数的交换子(compact)的性质，从而验证了Julg关于Baum-Connes猜想的剩余猜想。
We develop a generalization of quantitative $K$-theory, which we call controlled $K$-theory. It is powerful enough to study the $K$-theory of crossed product of $C^*$-algebras by action of étale groupoids and discrete quantum groups. In this article, we will use it to study groupoids crossed products. We define controlled assembly maps, which factorize the Baum-Connes assembly maps, and define the controlled Baum-Connes conjecture. We relate the controlled conjecture for groupoids to the classical conjecture, and to the coarse Baum-Connes conjecture. This allows to give applications to Coarse …
Given an action of a discrete quantum group (in the sense of Van Daele, Kustermans and Effros-Ruan) ${\cal A}$ on a $C^*$-algebra ${\cal C}$, satisfying some regularity assumptions resembling the proper $Γ$-compact action for a classical discrete group $Γ$ on some space, we are able to construct canonical maps $μ^i_r$ (\$μ_i$ respectively) ($i=0,1$) from the ${\cal A}$-equivariant K-homology groups $KK_i^{\cal A}({\cal C}, C |)$ to the K-theory groups $K_i(\hat{{\cal A}_r})$ ($K_i(\hat{\cal A})$ respectively), where $\hat{{\cal A}_r}$ and $\hat{\cal A}$ stand for the quantum analogues of the r…
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